Ananyo Dan, Inder Kaur
Let $K$ be the fraction field of a Henselian discrete valuation ring with algebraically closed residue field $k$. In this article we give a sufficient criterion for a projective variety over such a field to have index $1$.
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We prove a conjecture of L.Bonavero, C. Casagrande, O. Debarre and S. Druel, on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties.
Fano varieties with conjecturally largest Fano index
本文构造了具有典范和末端奇点的 Fano 簇中大 Fano 指标的示例，并基于低维证据猜想这些示例在所有维度上达到最大 Fano 指标。
Call a projective variety $X$ Calabi-Yau if its canonical divisor is ${\bf Q}$-linearly equivalent to zero. The smallest positive integer $m$ with $mK_X$ linearly equivalent to zero is called the index of $X$. We construct Calabi-Yau varieties with the largest known index in high dimensions. In our examples, the index grows doubly exponentially with dimension. We conjecture that our examples have the largest possible index, with supporting evidence in low dimensions. The examples are obtained by mirror symmetry from our Calabi-Yau varieties with an ample Weil divisor of small volume. We also g…
The period-index problem for hyperkähler varieties: Lower and upper bounds
本文通过证明超kähler变种的周期-指数猜想的最优性，并在K3^n型变种上验证了非特殊共素Brauer类的猜想。
Let $X$ be a Fano variety with at worst isolated quotient singularities. Our result asserts that if $C \cdot (-K_X) > max\{\frac{n}{2}+1,\frac{2n}{3}\}$ for every curve $C \subset X$, then $ρ_X=1$.
Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying this problem for Fano varieties of index one and two. The purpose of this paper is to give a new approach to this problem, and to show that in every positive characteristic there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphisms. To do this we prove that these varieties admit ample and birationally equivarian…
Let X be a Fano manifold of dimension n and index n-3. Kawamata proved the non vanishing of the global sections of the fundamental divisor in the case n=4. Moreover he proved that if Y is a general element of the fundamental system then Y has at most canonical singularities. We prove a generalization of this result in arbitrary dimension.
Around a class version of the Hodge index theorem for singular varieties
本文概述了奇异复代数簇霍奇指标定理特征类版本的最新进展，澄清了文献中L类概念的关系，并给出了该猜想成立的多种新情况，包括紧 toric 簇、Schubert 簇等。
We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety X with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibered by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.
On the rank index of some quadratic varieties
本文定义了射影簇齐次理想的秩指数，介绍了计算该指数的基本方法，并求得了有理正规卷积、del Pezzo簇、Segre簇及Grassmannian直线的Plücker嵌入等经典射影簇的秩指数值。
On a combinatorial description of the Gorenstein index for varieties with torus action
本文证明复杂度一的环面作用 Fano 簇及一般排列簇的 Gorenstein 指数可由反典范复形的格距离读出，并据此界定近齐次 Fano 三维簇的定义数据。
A conjecture of Pukhlikov states that a smooth Fano variety of dimension at least four and index one is birationally rigid. We show that a general member of the linear system given by the ample generator of the Picard group of the moduli space of stable, rank two bundles with fixed determinant of odd degree on a curve of genus at least three, is not birationally rigid.
Smooth Calabi-Yau varieties with large index and Betti numbers
本文构造了在每维中具有双指数增长指数的光滑射影Calabi-Yau变种，并探讨了其拓扑不变量的极端性质，认为这些变种在每维中可能是极值的。
On minimal log discrepancies on varieties with fixed Gorenstein index
本文推广了对数典范阈值聚点的有理性定理，并将其应用于极小对数差异的 ACC 问题，证明了固定 Gorenstein 指数簇上具有固定系数的三维典范对的极小对数差异满足 ACC。
This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have determinantal equations. In particular we establish the scheme theoretic version of a conjecture of Eisenbud-Koh-Stillman. If $L_1$ and $L_2$ are line bundles of degree at least $2g+1+k$, then ${\rm Sec}^j(C)$ is determinantally presented for $j \leq k$ and for $j \leq k+1$ if $L_1 \neq L_2$. We also give a geometric characterization of the standard Clifford index in t…
Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the Euler-Poincaré characteristic χ(X,E) if p=0 or p>dim(X)+1. If 0<p\leq dim(X)+1, the prime-to-p part of the index of X divides χ(X,E). Combining this with the Hattori-Stong theorem yields an analogous result concerning the divisibility of the cobordism class of X by the index of X. As a corollary, rationally connected varieties over the maximal unramified extension of a …
The period-index problem for hyper-Kähler varieties via hyperholomorphic bundles
本文通过射影超全纯丛，为$K3^{[n]}$型超Kähler簇的周期-指数问题建立了新界，并验证了Huybrechts猜想的部分情形。
We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension $3a$ and index $a+1$, for $a=1, 2, 4, 8$. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan's triality principle. We also prove that they are compactifications of affine spaces.
We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behaviour of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which resu…
We study log D-modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah's generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.
A local version of the arithmetic Hodge index theorem over quasiprojective varieties
本文提出拟射影簇上度量线丛的局部相交数，证明局部算术Hodge指标定理，并由此得到Monge-Ampère方程在阿基米德与非阿基米德情形下的唯一性。
An explicit bound for complements on toric varieties
该研究针对环面法诺簇上的klt补问题，推导得出其指标的有效上界，为代数几何中环面簇相关研究提供了显式的量化结果。
We study the canonical stability index of nonsingular projective varieties of general type with either large canonical volume or large geometric genus. As applications of a general extension theorem established in the first part, we prove some optimal results in dimensions 4 and 5, which are parallel to some well-known results on surfaces and 3-folds.
Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincaré-Hopf when the ambient variety is a smooth manifold. One has the Schwartz index, the local Euler obstruction, the GSV-index, the virtual index and the homological index (and probably other indices too). In this article we give a brief overview of all these indices and the relations amongst them. We also indicate their relations with the various generalisations to complex analytic singula…
Splitting unramified Brauer classes by abelian torsors and the period-index problem
本文利用扭曲相对Picard簇，通过固定阿贝尔概型的挠子分裂射影簇上的Brauer类，并证明非分歧Brauer类的指标整除其周期的固定幂次。
Minimal Trivializing Isogenies of $\mathbb{G}_m$-gerbes over Abelian Varieties and Period-Index Problem
本文通过引入新不变量 $Ind_{SH}(\alpha)$ 并研究 $\mathbb{G}_m$-gerbe 的 $\mu_n$-提升和 $1$-扭曲半齐次向量丛，改进了周期-指数关系，并证明了周期-指数猜想对任意维椭圆曲线乘积成立。
We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety $X$ and for any constructible function $α$ with respect to a complex algebraic Whitney stratification of $X$. We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of $α$. Whenever $α$ is the Euler obstruction of $X$, this index formula specializes to the Seade-Tibar-Verjovsky global counterpart of the Le-Teissier formula for the local Euler obstruction.
We use the Serre-Godeaux varieties of finite groups, projective representation theory, the twisted Atiyah-Segal completion theorem, and our previous work on the topological period-index problem to compute the etale index of Brauer classes alpha in some specific examples. In particular, these computations show that the etale index of alpha differs from the period of alpha in general. As an application, we compute the index of unramified classes in the function fields of high-dimensional Serre-Godeaux varieties in terms of projective representation theory.
We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of maximal flags indexed by Z_{>0}. We pay special attention to differences with the case of ordinary flag varieties.
We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where the key varieties are constructed by extending partially the Sarkisov…
We prove the existence of defective secant varieties of three-factor and four-factor Segre-Veronese varieties embedded in certain multi-degree. These defective secant varieties were previously unknown and are of importance in the classification of defective secant varieties of Segre-Veronese varieties with three or more factors.
We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their computation.
We give a characterization of Gorenstein toric Fano varieties of dimension $n$ with index $n$ among toric varieties. As an application, we give a strong version of Fujita's freeness conjecture and also give a simple proof of Fujita's very ampleness conjecture on Gorenstein toric varieties.
We prove a conjecture of Batryev which states that the family of all Fano varieties with kawamata log terminal singularities and fixed index, forms a bounded family.
Relations between indices of Calabi--Yau varieties and pairs
本文利用 Beauville–Bogomolov 分解进行维数归纳，证明光滑 Calabi–Yau 簇的指数可由更低维、带标准系数的 klt Calabi–Yau 配对实现。
In this work we provide effective bounds and classification results for rational $\QQ$-factorial Fano varieties with a complexity-one torus action and Picard number one depending on the invariants dimension and Picard index. This complements earlier work by Hausen, Süß and the author, where the case of free divisor class group of rank one was treated.
Index of coregularity zero log Calabi-Yau pairs
本文研究余正则性为0的对数卡拉比-丘对的指数，证明其Weil指数λ满足2λ(K_X+B)~0，给出精确上界，并推广到广义对数卡拉比-丘对、半对数典范对及孤立对数典范奇点等情形，还应用于Gross-Siebert纲领与带有限群作用的卡拉比-丘簇。
General linear sections of codimension 2 of the Grassmannians G(1,4) and G(1,5) appear in the classification of Fano manifolds of high index. Unlike Grassmannians, these manifolds are not homogeneous. Nevertheless, their automorphisms groups have finitely many orbits. In this work we first compute the orbits of these actions. Then we give a description of the variety of lines (under the Plücker embedding) passing through a fixed point in each orbit of the action. As an application we show that these Fano manifolds are not weakly 2-Fano, completing the classification of weakly 2-Fano manifolds …
In this article we will introduce, among others, the variety of subcomplexes and the variety of maps between complexes of given rank. Also, varieties of $\mathfrak{g}$-structure like $\mathfrak{g}$-Grassmannian, $\mathfrak{g}$-determinantal varieties and finally the variety $\mathcal{DGLA}(E)$ of differential graded Lie algebra structures on $E$. We will compute the dimensions of these varieties and also some relevant properties. The motivation of this article is to give examples of moduli spaces relevant to deformation theory.
This article is about 1-forms on complex analytic varieties and it is particularly relevant when the variety has non-isolated singularities. We first show how the radial extension technique of M.-H. Schwartz can be adapted to 1-forms, allowing us to define the Schwartz index of 1-forms with isolated singularities on singular varieties. Then we see how MacPherson's local Euler obstruction, adapted to 1-forms in general, relates to the Schwartz index, thus obtaining a proportionality theorem for these indices analogous to the one for vector fields. We also extend the definition of the GSV-index …
We give an example of a pair of projective symplectic varieties in arbitrarily large dimensions which are D-equivalent, L-equivalent, and birationally inequivalent.
We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperkähler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperkähler varieties.
We give various examples of Q-factorial projective toric varieties such that the sum of the squared torus invariant prime divisors is positive. We also determine the generators for the cone of effective $2$-cycles on a toric variety of Picard number two. This result is convenient to explain our examples.
Universal Brauer-Severi varieties
本文提出了一种通用布雷厄-塞弗里品种的构造方法，并研究其几何性质及上同调群，同时重新诠释了判别回避结果。
On Fano varieties of index one that are linear sections of Grassmannians of lines
本文研究线Grassmannian的一般线性截面Fano簇，证明其双有理等价于由pfaffian行列式定义的超曲面，并给出相应超曲面的有理参数化。
In this paper we show that the derived category of Brauer-Severi curves satisfies the Jordan-Hölder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field $k$. Moreover, we show that a $n$-dimensional Brauer-Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length $n+1$, at least in characteristic zero. We conjecture that Brauer-Severi varieti…
We study a wide class of affine varieties, which we call affine Fano varieties. By analogy with birationally super-rigid Fano varieties, we define super-rigidity for affine Fano varieties, and provide many examples and non-examples of super-rigid affine Fano varieties.
These are notes from talks of the authors on some explicit examples of families of Calabi-Yau threefolds which are parametrised by a Shimura variety. We briefly review the periods of Calabi-Yau threefolds and we discuss a recent result on Picard-Fuchs equations for threefolds which are hypersurfaces with many automorphisms. Next various examples of families parametrised by Shimura varieties are given. Most of these are due to J.C. Rohde. The examples with an automorphism of order three are given in some detail. We recall that such families do not have maximally unipotent monodromy and that the…
An old conjecture of Voisin describes how zero-cycles on a variety $X$ should behave when pulled-back to the self-product $X^m$ for $m$ larger than the geometric genus of $X$. Using complete intersections of quadrics, we give examples of varieties in any dimension and with arbitrarily high geometric genus that verify Voisin's conjecture.
In this article we study certain specialization properties of the index of varieties defined by Kollár.
Non-commutative hourglasses I: On classification of the Q-Fano 3-folds Gorenstein index 2 via Derived category
本文通过给出广义加权爆破的导出范畴公式，研究了Takagi簇在Sarkisov链接下的导出范畴变化行为，并利用非交换投影给出了例外收集及其几何意义。
We construct a family of examples of Legendrian subvarieties in some projective spaces. Although most of them are singular, a new example of smooth Legendrian variety in dimension 8 is in this family. The 8-fold has interesting properties: it is a compactification of the special linear group, a Fano manifold of index 5 and Picard number 1.
We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.
Complements and coregularity of Fano varieties
本研究探讨法诺簇的上正则性、对数Calabi-Yau对指标与补的关系，扩展了相关结果，证明不同绝对上正则性的法诺簇和klt奇点的补的存在性，并得到新典范丛公式与假设下的类似结论。
A Kawamata--Miyaoka type inequality for Fano varieties of arbitrary Picard number
针对维数不小于2的Q-可分典范弱Fano簇，证明当Q-Fano指数不小于3时其满足Kawamata–Miyaoka型不等式，并给出Gorenstein典范Fano三维簇的Q-Fano指数取值范围。
Total Cartier index of a bounded family
研究证明有界族的有理系数型正则型代数簇的总Cartier指数有界
Birational equivalence of Severi-Brauer varieties
本文证明了指数不是素数幂的 Severi-Brauer 簇的 Amitsur 猜想。
We study smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying with vanishing holomorphic Euler characteristic. We prove that the Albanese variety of $X$ has at least three simple factors. Examples were constructed by Ein and Lazarsfeld, and we prove that in dimension 3, these examples are (up to abelian étale covers) the only ones. By results of Ueno, another source of examples is provided by varieties $X$ of maximal Albanese dimension and of general type with $p_g(X)=1$. Examples were constructed by Chen and Hacon, and again, we prove that in di…
Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.
Generalize Kobayashi's example for the Noether inequality in dimension three, we provide examples of n-folds of general type with small volumes.
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